User blog:B1mb0w/Strong D Function
'Strong D Function' The strong D function is based on a weaker deeply nested Ackermann function called d. The rules are similar with the significant change being that the D function: \(D(x_1,x_2,x_3,x_4,...,x_n)\) expands to this function: \(D(x_1-1,D(x_1,x_2,x_3,x_4,...,x_n-1),...,D(x_1,x_2,x_3,x_4,...,x_n-1))\) The same expansion is used to replace each input parameter \(x_2\) to \(x_n\). Also refer to the Leading and Trailing Zero rules: L1 and T1, for more information. Definition For 2 parameters, the D function is equivalent to the d function: \(d(a,b)=d(a-1,d(a,b-1))=D(m,n)=D(m-1,D(m,n-1))\) For 3 parameters, the D function quickly dominates the weaker d function: \(d(a,b,c)=d(a-1,d(a,b-1),d(a,b,c-1))\) \(D(k,m,n)=D(k-1,D(k,m,n-1),D(k,m,n-1))\) 'Calculated Examples up to D(3,4)' \(D() = 0\) This is a null function that always returns zero. \(D(3) = 4\) This is the successor function \(D(1,2) = 5\) This is the same as d(1,2) \(D(2,3) = 17\) This is the same as d(2,3) \(D(3,9) = 1,240,025 >> 1,000,000\) \(D(3,206) = 122*10^{98} >>\) Googol \(D(3,4) = 5099 >> f_2(6) = 6.2^6 = 384\) Calculated Examples up to D(1,0,n) Using the Comparison Rule C1 \(d(m,n-2+\delta) >> f_{m-1}(n)\) where \(\delta << n\) \(D(4,1) >> f_{3}(3) = f_{\omega}(3)\) \(D(4,2) >>\) Googolplex \(D(4,4) >> f_{3}^3(f_{\omega}(3))\) \(D(6,1) >> g_1\) where \(g_{64} = G\) Graham's number \(D(D(4,1),D(4,1)) >> D(f_{\omega}(3)+1,f_{\omega}(3)-2) >> f_{\omega}(f_{\omega}(3)) = f_{\omega}^2(3)\) \(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(4,4) >> f_3(6) >> f_{\omega}(3)\) \(D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >> f_{\omega}(f_3(6))\) \(D(1,0,2) >> f_{\omega}^2(f_3(6)) >> f_{\omega}^2(f_3(3)) >> f_{\omega}^2(f_{\omega}(3)) >> f_{\omega}^3(3) >> f_{\omega+1}(3)\) \(D(1,0,3) >> f_{\omega}^3(f_3(6))\) \(D(1,0,4) >> f_{\omega}^4(f_3(6)) >> f_{\omega}^4(4) >> f_{\omega+1}(4)\) and \(D(1,0,n+\delta) >> f_{\omega}^n(n) = f_{\omega+1}(n)\) where \(\delta << n\) 'Calculated Example for Graham's number' \(D(1,1,0) = D(0,D(1,0,1),D(1,0,1)) = D(1,0,2)\) \(D(1,1,1) = D(0,D(1,1,0),D(1,1,0)) = D(0,D(1,0,2),D(1,0,2)) = D(1,0,3)\) \(D(1,1,n) = D(1,0,n+2)\) \(D(1,2,0) = D(0,D(1,1,2),D(1,1,2)) = D(1,1,3) = D(1,0,5)\) \(D(1,2,n) = D(1,1,n+3) = D(1,0,n+5)\) \(D(1,3,n) = D(1,2,n+4) = D(1,1,n+7) = D(1,0,n+9)\) then \(D(1,m,n) = D(1,m-1,n+m+1) = ... = D(1,0,n-1+(m+2).(m+1)/2)\) and \(D(1,9,9) = D(1,6,36) = D(1,3,54) = D(1,0,63) >> g_{64} = G\) Graham's number Calculated Examples up to D(2,0,n) Using \(D(1,0,n-1+(m+2).(m+1)/2)\) we get \(D(1,2,2) = D(1,0,1+4.3/2) = D(1,0,7) >> f_{\omega+1}(7)\) then \(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >> D(1,0,D(1,2,2)) >> f_{\omega+1}^2(7)\) and \(D(2,0,1) = D(2,D(2,0,0),D(2,0,0)) >> f_{\omega+1}^3(7) >> f_{\omega+1}^3(3) = f_{\omega+2}(3)\) \(D(2,0,2) >> f_{\omega+1}^4(7) >> f_{\omega+1}^4(4) = f_{\omega+2}(4)\) \(D(2,0,3) >> f_{\omega+1}^5(7) >> f_{\omega+1}^5(5) = f_{\omega+2}(5)\) and \(D(2,0,n-2) >> f_{\omega+2}(n)\) when \(n < 7\) or \(D(2,0,n-2+delta) >> f_{\omega+2}(n)\) when \(\delta << n\) Calculated Examples up to D(n,0,n) \(D(2,3,3) = D(2,0,2+5.4/2) = D(2,0,12) >> f_{\omega+2}(13)\) then \(D(3,0,0) = D(2,D(2,3,3),D(2,3,3)) >> D(2,0,D(2,3,3)) >> f_{\omega+2}^2(13)\) and \(D(3,0,1) >> f_{\omega+2}^3(13) >> f_{\omega+3}(3) = f_{\omega.2}(3)\) \(D(3,0,2) >> f_{\omega+2}^4(13) >> f_{\omega+3}(4)\) and \(D(3,0,n-2) >> f_{\omega+3}(n)\) when \(n < 13\) or \(D(3,0,n-2+\delta) >> f_{\omega+3}(n)\) when \(\delta << n\) then \(D(4,0,n-2+\delta) >> f_{\omega+4}(n)\) and \(D(n,0,n-2+\delta) >> f_{\omega+n}(n) = f_{\omega.2}(n)\) where \(\delta << n\) Calculated Examples up to D(n,0,0,n) \(D(1,0,0,0) = D(0,D(0,1,1,1),D(0,1,1,1),D(0,1,1,1))\) \(= D(D(1,1,1),D(1,1,1),D(1,1,1)) >> f_{\omega.2}(D(1,1,1)) = f_{\omega.2}(D(1,0,3)) >> f_{\omega.2}(f_{\omega}^3(f_3(6)))\) \(D(1,0,0,1) >> f_{\omega.2}^2(n)\) where \(n <= f_{\omega}^3(f_3(6))\) then \(D(1,0,0,n-1+\delta) >> f_{\omega.2}^n(n) >> f_{\omega.2+1}(n)\) where \(\delta << n\) \(D(2,0,0,n-2+\delta) >> f_{\omega.2+2}(n)\) where \(\delta << n\) \(D(3,0,0,n-2+\delta) >> f_{\omega.2+3}(n)\) where \(\delta << n\) or \(f_{\omega^2}(3)\) when \(n = 3\) and \(D(n,0,0,n) >> f_{\omega.2+n}(n) >> f_{\omega.3}(n)\) where \(\delta << n\) 'Strong D Function Growth Rate' Using subscript notation \(0_{n-1}\) to represent n-1 input parameters with value 0: \(D(1,0,0,0,n) = D(1,0_{3},n) >> f_{\omega.3+1}(n)\) \(D(n,0_{3},n) >> f_{\omega.4}(n)\) or \(f_{\omega^2}(4)\) when \(n = 4\) then \(D(n,0_{4},n) >> f_{\omega.5}(n)\) or \(f_{\omega^2}(5)\) when \(n = 5\) and \(D(n,0_{n-1},n) >> f_{\omega.n}(n) = f_{\omega^2}(n)\) 'Some calculations for n=2' \(D(2,0) = 8 = f_2(2) = f_{\omega}(2)\) \(D(2,0,2) >> D(2,0,0) >> f_{\omega^2}(2) = f_{\omega^{\omega}}(2) = f_{\epsilon_0}(2)\) 'Some calculations for n=3' \(D(4,1) >> f_{\omega}(3)\) \(D(1,0,2) >> f_{\omega+1}(3)\) \(D(2,0,1) >> f_{\omega+2}(3)\) \(D(3,0,1) >> f_{\omega.2}(3)\) \(D(3,0_{2},1) = D(3,0,0,1) >> f_{\omega^2}(3)\) \(D(D(3,0_{2},1),0_{[D(3,0_{2},1)]},1) >> f_{\omega^2}^2(3)\) 'Next - the Alpha Function' My next blog post will introduce a new Alpha function that I have been thinking about. 'References' Strong D Function *Deeply Nested Ackermann Function **Modified Ackermann Function The following references are outdated because the above proofs and examples are more reliable. Please keep this in mind if you refer to any of these blogs. *(Outdated) References **Summary of Strong D Function Growth Rates ***General Proof of Strong D Function Growth Rate **Strong D Function and f epsilon nought (n)